Quantization via Linear homotopy types
Abstract
In the foundational logical framework of homotopytype theory we discuss a natural formalization of secondary integral transforms in stable geometric homotopy theory. We observe that this yields a process of nonperturbative cohomological quantization of local prequantum field theory; and show that quantum anomaly cancellation amounts to realizing this as the boundary of a field theory that is given by genuine (primary) integral transforms, hence by linear polynomial functors. Recalling that traditional linear logic has semantics in symmetric monoidal categories and serves to formalize quantum mechanics, what we consider is its refinement to linear homotopytype theory with semantics in stable infinitycategories of bundles of stable homotopy types (generalized cohomology theories) formalizing Lagrangian quantum field theory, following Nuiten and closely related to recent work by Haugseng and HopkinsLurie. For the reader interested in technical problems of quantization we provide nonperturbative quantization of Poisson manifolds and of the superstring; and find insight into quantum anomaly cancellation, the holographic principle and motivic structures in quantization. For the reader inclined to the interpretation of quantum mechanics we exhibit quantum superposition and interference as existential quantification in linear homotopytype theory. For the reader inclined to foundations we provide a refinement of the proposal by Lawvere for a formal foundation of physics, lifted from classical continuum mechanics to local Lagrangian quantum gauge field theory.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 DOI:
 10.48550/arXiv.1402.7041
 arXiv:
 arXiv:1402.7041
 Bibcode:
 2014arXiv1402.7041S
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Mathematics  Algebraic Topology;
 Mathematics  Logic;
 0306;
 18F99;
 1906;
 5502;
 8102
 EPrint:
 89 pages